US soldier kills Afghan civilians in Kandahar
The White House voiced "deep concern" and Nato-led forces in Afghanistan promised a rapid inquiry.
Afghan President Hamid Karzai has condemned the attack and demanded an explanation from Washington.
BBC correspondents say there could be a furious backlash when news of the attack reaches the wider public.
In Kandahar's Panjwai district, local people have gathered near the base to protest about Sunday's killings, and the US embassy is advising against travel to the area.
Anti-US sentiment is already high in Afghanistan after US soldiers burnt copies of the Koran last month.
US officials have apologised repeatedly for the incident at a Nato base in Kabul but they failed to quell a series of protests and attacks that killed at least 30 people and six US troops.
'Shot and burnt'
The unnamed soldier, thought to be a staff sergeant, is reported to have walked off his base at around 03:00 local time (22:30 GMT Saturday).
AnalysisThe US embassy in Kabul is warning of possible anti-American reprisals. These latest killings can only play into the already severely strained relationship between the US and Afghanistan after US soldiers burned copies of the Koran at a Nato military base near Kabul.
That incident sparked violent anti-American demonstrations. Britain's ambassador to Afghanistan has said that the latest incident is a "unique tragedy" and should not affect the timetable for the withdrawal of British and American combat troops by the end of 2014.
But Washington and Kabul are currently negotiating a long-term strategic partnership which will govern their relations from 2014 onwards. President Karzai was delivering a speech about the transition as news of today's incident started to come in.
He stressed that any international forces remaining in the country then for training and other purposes would have to operate under strict guidelines regarding their responsibilities and when they could leave their bases.
"When it was happening in the middle of the night, we were inside our houses," he said. "I heard gunshots and then silence and then gunshots again."
In one house in Najeeban, the gunman reportedly killed 11 people, setting fire to their bodies before he left.
A relative of the 11 victims, Haji Samad, told Reuters news agency chemicals had been poured over the bodies and set alight.
"I saw that all 11 of my relatives were killed, including my children and grandchildren," he added, weeping.
An unnamed woman witness in Najeeban told BBC News she had heard gunfire at about 02:00. A barking dog was shot dead by the gunman, she said.
She added that the Taliban had not been seen in the area for five months.
At least three of the child victims were killed by a single shot to the head, the BBC's Quentin Sommerville reports from Kabul.
Photographs from the scene showed bodies, some of them clearly young children, placed in a vehicle under blankets.
Five wounded Afghans were taken to Nato medical facilities, the Associated Press reports.
Some sources suggested that more than one soldier was involved in the attack, and a statement by the Taliban accused Afghan security forces of playing a role.
A delegation from the provincial governor's office has arrived in the village to determine exactly what happened, a spokesman said.
The soldier - who reportedly suffered a breakdown before the attacks - is said to have handed himself over to the US military authorities after carrying out the killings.
In a statement, Mr Karzai described the deaths in Kandahar as "intentional murders".
Previous tension points
- February 2012: Violent protests erupt after US troops inadvertently burn copies of the Koran at Bagram air base in Kabul. At least 30 are killed.
- January 2012: US and UN officials describe a video clip of US marines urinating on dead Afghans as "disgusting" and "inhuman"
- April 2011: US President Barack Obama describes March 2011 Koran burning by a radical US pastor as "intolerance and bigotry". The incident triggered protests which left at least 24 people dead in Afghanistan
- April 2008: Dutch and Danish governments evacuate their embassies in Kabul after protests against cartoon of the Prophet Muhammad which was reprinted by Danish newspapers
Caitlin Hayden, a spokeswoman for the White House National Security Council, said US President Barack Obama had been briefed on the incident. She added: "We are deeply concerned by the initial reports of this incident, and are monitoring the situation closely."
Gen John R Allen, commander of the Nato-led International Security Assistance Force (Isaf), said US officials in Afghanistan would work with their Afghan counterparts to investigate what had happened.
"I pledge to all the noble people of Afghanistan my commitment to a rapid and thorough investigation," he said in a statement.
This is the first time Afghan civilians have been targeted by foreign soldiers in this way, our Kabul correspondent says.
However, a US soldier was convicted last year on three counts of premeditated murder after leading a rogue "kill team" in Afghanistan.
Kandahar is the Taliban's spiritual heartland and is considered strategically important because of its international airport, its agricultural and industrial output and its position as one of the country's main trading hubs.
The province has seen heavy fighting between Nato and Taliban forces over the last five years.
Mr Karzai said earlier he still expected to sign a strategic partnership with the US in the next couple of months.
He said discussions would continue on the precise role the US would play in Afghanistan after Nato handed over security responsibility to Kabul at the end of 2014.
On Friday, Kabul and Washington reached a deal to transfer US-run prisons in the country to Afghan control.